begin

    using Plots

    using Images

    using LinearAlgebra

end

begin

    img = load("../data/dog.jpg")

    X = convert(Array{Float64}, Gray.(img))

    plot(img)

end

begin

    (U, Σ, V) = svd(X, full=false)

    S = Diagonal(Σ)

    Vt = V'

    rs = [length(Σ), 5, 20, 100]

    p_1 = plot(layout=length(rs), size=(600, 600))

    for i ∈ 1:length(rs)

        r = rs[i]

        Xapprox = U[:, 1:r] * S[1:r, 1:r] * Vt[1:r, :]

        plot!(p_1[i], Gray.(Xapprox), axis=nothing)

        percentage = round(r*100/length(Σ), digits=2)

        title!(p_1[i], "r=$(r) $(percentage)%")

    end

    p_1

end

begin

    p_2 = plot(layout=2, legend=false)

    plot!(p_2[1], Σ, yaxis=:log)

    title!(p_2[1], "Singular Values")

    plot!(p_2[2], cumsum(Σ) ./ sum(Σ))

    title!(p_2[2], "Singular Values:\nCumulative Sum")

    p_2

end